In the Hermitian case, the QGT is the sum of a symmetric tensor, known as the quantum metric, plus an anti-symmetric part proportional to the Berry curvature [1]}. We find a similar decomposition here. The anti-symmetric part of \(\mathcal {G}\)  is readily seen to be \(K/2\) , as
\(\mathcal {G}^{\mathrm {alt}}_{(A,v,\theta )}(\mathfrak {v}_1,\mathfrak {v}_2)=\theta \frac{[X_1,X_2]}{2}v=\frac{1}{2}\mathrm {d}\theta \wedge \mathrm {d}v(\mathfrak {v}_1,\mathfrak {v}_2)=\frac{1}{2}K_{(A,v,\theta )}(\mathfrak {v}_1,\mathfrak {v}_2).\) 
